1. Introduction
Water vapor supersaturation is a key thermodynamic parameter in the formation and development of warm and cold clouds. The activation of cloud condensation nuclei (CCN), the spectrum of droplet sizes, and therefore, precipitation heavily depend on the supersaturation budget of an evolving cloud. By extension, the study of supersaturation is fundamental to determine the radiative properties of cloud fields which constitute a major climate feedback (Khain and Pinsky 2018), yet a major source of uncertainty in climate projections (Zelinka et al. 2020).
Due to its microphysical character, supersaturation cannot be explicitly resolved by global circulation models (GCMs), and hence, it has to be prescribed using adequate parameterization schemes. One way of doing this is by predicting the probability of encountering a cloud parcel at a certain relative humidity level in terms of the background dynamics and large-scale information. Taking the GCM perspective, the idea is to infer the subgrid cloud properties out of the prognosed variables: vertical velocity, temperature, specific humidity, and pressure, which altogether depict the relative humidity configuration of a homogeneous cloud parcel. However, localized temperature gradients and turbulence at the smaller scales can create inhomogeneous fluctuations which alter the spread of supersaturation values, hence changing the microphysical properties of the subgrid cloud parcel.
Turbulence, in fact, influences the life cycle of clouds, from the smallest warm cumuli to the coldest stratiform clouds (Shaw 2003). Indeed, the incorporation of turbulent models into the determination of microphysical properties has played a key role in explaining the broadening of droplet size spectra in developing cumulus clouds (Grabowski and Abade 2017; Abade et al. 2018) and mixed-phase clouds (Hoffmann 2020; Chen et al. 2023), as well as in the activation and maintenance of long-lasting supercooled liquid water in icy clouds (Korolev and Field 2008; Hoffmann 2020). Moreover, laboratory experiments suggest that turbulence at the smallest scales are also responsible for CCN activation even in mean-subsaturated environments (Prabhakaran et al. 2020).
It is crucial, then, to determine not only the mean value of supersaturation at a given location, but also its variance or even higher moments. To this end, the equations for supersaturation or condensational growth are coupled with a suitable stochastic forcing law that captures the effects of turbulence affecting both vertical velocity fluctuations (Bartlett and Jonas 1972; Sardina et al. 2015) as well as the integral radius of the droplet distribution (Manton 1979; Cooper 1989). Hence, the classical deterministic models become stochastic differential equations (SDEs), which have been widely used in the cloud physics community; see, e.g., McGraw and Liu (2006), Paoli and Shariff (2009), Sardina et al. (2015), and Siebert and Shaw (2017). In fact, this stochastic physics framework has been employed in the elaboration of analytically tractable parameterization schemes for mixed-phase clouds (Furtado et al. 2016), the study of droplet growth by condensation (Bartlett and Jonas 1972; Sardina et al. 2015), and the determination of steady-state warm cloud properties (Siewert et al. 2017).
The supersaturation equation, also known as Squires equation (Squires 1952), consists of a nonlinear relation between supersaturation with its sources and sinks, and has extensively been studied; see, e.g., Korolev and Mazin (2003) or Devenish et al. (2016)—we also refer to Eq. (2) of this paper. The addition of turbulent updrafts modeled by stochastic processes into the linearized Squires equation allowed to obtain Gaussian formulas for the ice-supersaturation (probability) distribution (Field et al. 2014), which serves to diagnose the amount of supercooled liquid water in mixed-phase clouds (Furtado et al. 2016). Along these lines, Sardina et al. (2015) also provided a relation for the moments of supersaturation and droplet size distribution. However, some restrictions were imposed in these works. First, the full, nonlinear form of Squires equation has never been examined in the stochastic context, unlike its deterministic counterpart [Korolev and Mazin 2003, Eqs. (9) and (10)]. In fact, this nonlinear form is specially relevant for clouds experiencing high maximum supersaturation, as pointed out in Devenish et al. (2016). Second, in the limit of fast turbulent decorrelation—relative to supersaturation time scales—the stochastic turbulence becomes white noise, which can be a strong assumption if one is concerned with other shorter relevant time scales like that at the CCN activation level, as already warned by Field et al. (2014) and Abade et al. (2018). Third, the time invariance of mean droplet radius—the quasi-steady assumption (Squires 1952)—appears critical to obtain analytical supersaturation distributions (Field et al. 2014; Furtado et al. 2016). Turbulence, mixing, and entrainment, however, can provoke fluctuations in droplet integral radius (Manton 1979; Cooper 1989), which consequently perturb the supersaturation budget. This work is concerned with analytically assessing and lifting the listed assumptions in the study of the supersaturation equation.
This paper is structured as follows. In section 2, the Squires equation for the evolution of supersaturation is derived, and second, a stochastic equation for turbulent updrafts is presented, in the lines of the theory of stochastic Lagrangian turbulent models; see, e.g., the work of Rodean (1996). In section 3, the quasi-steady equation—which assumes a constant mean droplet/particle radius—is analyzed in a number of approximations providing new formulas for the probability distribution of supersaturation. In section 4, fluctuations in droplet size are allowed, and their net effects on supersaturation distribution are investigated in analytical terms. A total of five different supersaturation distributions are derived. In section 5, the relevance of the five obtained probability density functions is discussed and compared in the context of mixed-phase clouds: we compare the supercooled liquid cloud fraction and liquid water contents predicted by each distribution. Finally, a discussion over the results is done in section 6. To supplement the information in the main text, appendixes are included to discuss some technical topics related to the analysis of stochastic differential equations.
2. The supersaturation equation
a. The Squires equation
We consider a vertically moving cloud parcel containing a monodisperse family of liquid water droplets or ice particles—allowing mixed-phase conditions—that are spatially uniformly distributed. Furthermore, it is assumed that the number of droplets/particles per unit mass is constant in time. The evolution of supersaturation in the cloud is, essentially, determined by the sources and sinks of relative humidity due to the adiabatic cooling of the parcel in ascent, and condensation of water vapor onto the existing droplets’ or particles’ surface (Rogers and Yau 1989). However, the exact relation between the rate of change of supersaturation and its sinks and sources is obtained by taking its derivative with respect to time.
b. Stochastic model for updrafts
3. Quasi-steady model statistics
a. Fast Lagrangian decorrelation time scale
Phase relaxation to equilibrium. The blue curve represents the average of 10 ensemble means of 103 noise realizations of Eq. (19) for an icy cloud, with a prescribed mean updraft of
Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1
Linear approximation
b. Slow Lagrangian decorrelation time scale
Supersaturation variance as a function of TKE. The blue colors correspond to the variances for the 2D linear system (30) with red-noise updrafts, whereas the black colors correspond to the homogenized 1D Eq. (26), where updrafts become white in time. The dots are obtained by calculating the variance of 106-s time series with a time step of 10−2 s for each value of TKE. The solid lines are the predicted variance using Eqs. (22a) and (35), for the blue and orange curves, respectively. We highlight that when turbulence is less energetic, both estimates become more similar. The ambient conditions in this numerical experiment are SE = 0,
Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1
4. Diffusional growth and size fluctuations
In the section 2 we presented the full and closed equation for the evolution of supersaturation accounting for the condensational (depositional) growth of droplets (ice crystals) coupled to turbulent updraft fluctuations; see Eq. (6). In section 3, we investigated the properties of the stochastic Squires equation, describing the evolution of supersaturation in the quasi-steady approximation, where condensational growth is neglected. Here, we aim at lifting the quasi-steady assumption, by coupling supersaturation to condensational growth. In this regard, analytical work is limited because of the complexity of the resulting equations (Korolev and Mazin 2003). Nevertheless, approximations have been made to calculate the solutions of the supersaturation equation for small times (Devenish et al. 2016), and to determine the linear growth of variance of droplet radius as time increases (Sardina et al. 2015). In this section we shall provide, first, a brief review of the problem of droplet radius variance increase under random updrafts and, second, present a modified supersaturation equation that accounts for random fluctuations in droplet radius. The novel results are two different probability distributions for supersaturation in the spirit of the previous sections.
While under this framework there is no stationary distribution with finite variance for particle radius, it was shown in Siewert et al. (2017) that if the ambient supersaturation SE is negative, i.e., subsaturated, the probability distribution of r2 will possess the structure of an exponential function with a Dirac peak located at r2 = 0, which arises from the boundary condition in Eq. (36c). Indeed, it is expected that the trajectories in the S–r2 plane of Eq. (36) will display cycles, where r2 grows but then vanishes and sticks at the boundary of r2 = 0 for an open interval of time; see Fig. 4 of Siewert et al. (2017).
a. Fluctuations in droplet radius
In the previous section we clarified that if supersaturation is let to be driven by random turbulent updrafts, the mean-square radius grows linearly in time, and therefore, unbounded Brownian excursions can be expected when solving the condensational growth equation. This implies that variance is not bounded and steady-state distributions of droplet radius have infinite variance. In this section, the target is to compromise between the quasi-steady approximations of section 3 and the coupling with diffusional growth as done in Eq. (36). This is done by accounting for variability in integral radius due to mixing with the cloud’s exterior, perturbations in the particles’ capacitance—particularly for ice particles; see Eq. (4)—and small thermal fluctuations that affect the equilibrium vapor pressure at each droplet’s surface. Small radius fluctuations were already theoretically suggested in the work of Manton (1979) and further explored in Cooper (1989), where the authors consider their effect in the broadening of the droplet size spectrum. Under the quasi-steady approximation, it is concluded that updraft fluctuations alone cannot provoke a broadening of droplet size spectra, but turbulent fluctuations in the integral radius have to be considered; see Eq. (10) in Cooper (1989) and associated comments.
b. Supersaturation distribution comparison
At this stage we overview the qualitative mathematical features of the calculated supersaturation distributions
Comparison between PDFs. The legend indicates the analytically obtained PDFs
Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1
5. Calculating mixed-phase cloud properties
Mixed-phase clouds constitute a large portion of the global cloud coverage. Altocumulus, altostratus, stratocumulus, and Arctic stratus are instances where mixed-phase conditions have been observed to exist and persist at temperatures as low as −40°C; see, e.g., Korolev et al. (2017). The coexistence of liquid water and ice is, however, thermodynamically unstable so that, in freezing temperatures, ice will inevitably grow at the expense of liquid water (Bergeron 1935). It was suggested in Korolev and Field (2008) a mechanism by which supercooled liquid water can be activated in an initially icy cloud. According to such work, liquid water is activated in an icy cloud if the following two criteria are met: (i) the vertical velocity must exceed a threshold value and (ii) the cloud parcel must be lifted to a threshold altitude. As a consequence, dynamical forcing is responsible for the production of supercooled liquid water and, moreover, different updraft profiles yield different mixed-phase conditions.
It was already advanced in Korolev and Field (2008) that turbulent updrafts could be a mechanism to produced mixed-phase conditions faithful to what is observed, for example, in stratiform clouds; see also Li et al. (2019) for a study on the effects of supersaturation fluctuations on droplet growth on such clouds. Indeed, supersaturation fluctuations can provoke liquid-water saturation conditions in icy parcels, which translates to the activation of supercooled liquid water. In addition, numerical simulations indicate that small-scale turbulent mixing decelerates the ice growth and, thus, extends the lifetime of supercooled water (Hoffmann 2020). In Field et al. (2014), an analytical framework was established for determining mixed-phase properties for icy clouds in turbulent environments. Such framework is based on the analysis of the (ice) supersaturation equation subject to the linear approximation and random turbulent updrafts modeled by white noise; this is revisited in section 3a(1).
The five probability distributions obtained in the previous sections are now used to compute the supercooled liquid cloud fraction and the mean LWC of an adiabatic cloud parcel for a range of free parameters. Such free parameters are the TKE, as a proxy for turbulent forcing, and variance of the droplet radius fluctuations. In Fig. 4 we show the dependence of the mentioned partial moments on values of TKE at two temperatures indicated in the captions: Figs. 4a and 4b show supercooled liquid cloud fraction and mean LWC at −10°C, whereas Figs. 4c and 4d show the same at −5°C. Such statistics where computed using a simple quadrature scheme on the interval [−10, 10] so that all the considered PDFs integrate to unity with a tolerance of 10−10. We observe a monotone dependence on TKE in all the PDFs but for the gamma distribution, which yielded decreasing cloud fractions for values of TKE > 6 m2 s−2 at −10°C and TKE > 4 m2 s−2 at −5°C. Such change in trend is due to the displacement of the mode and tail thickness in the gamma distribution as the location factor is altered due to the multiplicative noise. It is noted that the red and orange curves, for f2 and f4, respectively, are almost coinciding.
(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of TKE. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against TKE for each analytical supersaturation distribution
Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1
When droplet size fluctuations are allowed to vary, the first three calculated distributions f1, f2, and f3 naturally yield the same statistics for supercooled liquid cloud fraction and mean LWC. Contrarily, when noisy variations in radius are allowed the distributions of f4 and f5 are likely to display a dependence on σr. This dependence is shown in Fig. 5, where the supercooled liquid cloud fraction and LWC are calculated as a function σr, for f2, f4, and f5. Figures 5a and 5b show the results at −10°C, whereas Figs. 5c and 5d show the same at −5°C. Because f4 and f5 are expensive to evaluate at small values of σr, ergodic averages are computed instead, i.e., the first equality in Eqs. (45) and (47). Such averages are taken over an integration of Eq. (40) over 106 seconds. While f2 is expectedly constant, f4 and f5 start to be dependent for values larger than
(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of σr. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against σr for the analytical distributions f1, f4, and f5 and for two temperature configurations. For small values of σr, the functions f4 and f5 are computationally expensive to evaluate so ergodic averages of 106 seconds are taken instead. In (a) and (b), a temperature of −10°C is considered, while in (b) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are SE = 0,
Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1
6. Discussion
In this paper, the analysis of the stochastic Squires equation has been done, with the aim of providing analytical formulas for the distribution of supersaturation in an adiabatic cloud parcel. Such equation describes the evolution of supersaturation over time by, essentially, taking into account the sources and sinks of relative humidity due to (i) adiabatic cooling via turbulent updrafts and (ii) water vapor condensation onto droplets/particles. Turbulent effects on updrafts and droplet/particle radius are here modeled as a red-noise process, in accordance with the spatial and time correlations of isotropic fluctuations in the inertial regime (Rodean 1996). Thus, we presented a hierarchy of five supersaturation equations that, by using the Fokker–Planck equation, possess analytically tractable distributions whose properties were assessed in the context of mixed phase clouds.
The theoretical results of Field et al. (2014)—here condensed in section 3a(1)—have been here generalized to a wider range of contexts. First of all, the Squires equation is considered in section 3 in its full nonlinear version and has been shown to possess a gamma-like stationary distribution—here denoted as f1—that deviates from a Gaussian according to the parameter α defined in Eq. (22a). Such parameter, also found in Field et al. (2014), is a nondimensional ratio between the turbulent fluctuation time scales and the phase relaxation coefficient. Thus, as turbulent fluctuations decrease in variance (relative to the microphysical or mixing time scale), f1 becomes better and better approximated by the Gaussian distribution f2. Indeed, this can be seen in section 5 from the computation of partial moments for mixed-phase clouds—mixed cloud fraction and mean LWC—in Fig. 4.
The time-scale separation assumption between updraft fluctuations and supersaturation, needed to compute f1 and f2, is lifted in section 3b. Indeed, f1 and f2 are only valid when supersaturation time scales are much greater that the Lagrangian decorrelation τd so that updraft fluctuations become uncorrelated in time; cf. Pavliotis (2014, chapter 11, result 11.1). The resulting distribution f3 is still Gaussian, but possesses a modification in the variance compared to that of f2; see Eq. (35). In Fig. 2, we show that the variance predicted by f2 and f3 diverge as TKE increases.
The main assumption needed to compute f1, f2, and f3 is the quasi-steady approximation, whereby the droplet or ice particle radius is considered constant. When coupling droplet growth and supersaturation in a turbulent environment, it was shown in Sardina et al. (2015) that the long-term variance of droplet squared radius scales linearly in time, similar to Brownian motion. This is revisited here in section 4. We highlight that such result is only valid for short times, since large families of droplets are subject to processes like sedimentation, evaporation, or mixing with exterior dry air that provoke a memory loss in collective droplet growth, and hence, the variance formula is reinitialized. Here, we proposed a modification of the supersaturation equation where the sink term—which depends on the integral radius of the droplet/particle population—is allowed to fluctuate randomly, as suggested in earlier work (Manton 1979; Cooper 1989). Two situations are studied: (i) fluctuations in updraft are uncorrelated to those of droplet size and (ii) the source of noise is the same, albeit with different intensities. The resulting model yields analytically tractable probability distributions for supersaturation. The net effect of droplet radius fluctuations is illustrated in Fig. 5, where the supercooled liquid cloud fraction and mean LWC are computed as a function of σr. It is found that correlated radius and updraft fluctuations yield a probability distribution that deviates severely from the quasi-steady approximation by up to +5% in supercooled liquid cloud fraction when σr ≥ 10−6 m. Contrarily, the mean LWC is negatively correlated with fluctuations in σr in case of f4. Regarding the uncorrelated sources of noise, the supercooled liquid cloud fraction appears to be weakly dependent in σr. On the other hand, mean LWC correlates positively is droplet size fluctuation variance. All the obtained PDFs are also qualitatively compared in Fig. 3.
There is still a need to provide a more quantitative verification of the formulas here presented. However, the formulas and discussions in this paper suggest that the widely used linear approximation for supersaturation evolution—see Eq. (26)—is limited to clouds experiencing low supersaturation, where the quasi-steady approximation is valid and updrafts decorrelate instantly with respect to phase relaxation time scales. The mixed-phase cloud stochastic parameterization scheme developed in Furtado et al. (2016) is based on these assumptions, and hence, future work should be oriented toward discerning which specific atmospheric conditions are appropriate for each of the here calculated supersaturation PDFs
On a more theoretical note, a deeper investigation of Eq. (6) would be of great interest in this stochastic framework. One step forward would be to impose a characteristic time for the loss of memory, so that the integrodifferential equation can be replaced by a simpler expression, possibly some form of noise with suitable time-decorrelation properties. We anticipate that this would entail technical difficulties due to the square root nonlinearity—here minimally tackled in appendix C—so research should be oriented in this direction. In general, we belief that this statistical-physics approach can be extended to more general contexts, possibly, by including more microphysical processes that affect the growth of liquid droplets or ice particles and, hence, the overall regulation of a cloud’s supersaturation budget.
Acknowledgments.
The authors thank S. Roncoroni, P. Field, and B. Devenish for their comments, suggestions, and kind reception at the Met Office. MSG is grateful to the Mathematics of Planet Earth Centre for Doctoral Training (MPE CDT) for making this collaboration possible. MSG acknowledges and is grateful for the support of the Institute of Mathematics and its Applications (Grant Number SGS21/08). MSG is thankful to I. Koren, M. D. Chekroun, the cloud physics group, and the graduate school at the Weizmann Institute of Science for providing a most inspiring environment.
Data availability statement.
No data have been used in this publication apart from the numerical integration of the equations here presented. The numerical schemes employed are found in https://doi.org/10.5281/zenodo.7904642.
APPENDIX A
List of Some Used Notations and Symbols
Table A1 provides a list of symbols that appear in the main text.
List of symbols.
APPENDIX B
Stationary Supersaturation Distribution
The general case
APPENDIX C
The Square Root Process
To support this analytical expansion, we numerically sampled an adimensional random variable X, normally distributed with mean 2 and standard deviation σX, where the latter takes 250 equispaced values between 10−4 and 0.5. The sample is of size 105 draws. With this set of data, we are able to numerically estimate the mean and variance of the square root random variable,
Variance and mean of the square root random variable. The black dots are calculated as follows: for each value of σX, the variance and mean of the random variable
Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1
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